VECTOR

S

A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

Two vectors are equal if they have the same direction and magnitude (length).

Blue and orange vectors have same magnitude but different direction.

Blue and green vectors have same direction but different magnitude.

Blue and purple vectors have same magnitude and direction so they are equal.

Component Form of a Vector

The component form of the vector with initial point P = (p1, p2)and terminal point Q = (q1, q2) is

PQ vvvpqpq 212211 ,,

The magnitude (or length) of v is given by

222

211 pqpqv 2

22

1 vv

Find the component form and length of the vector v that hasinitial point (4,-7) and terminal point (-1,5)

-2 2 4

6

4

2

-2

-4

-6

-8

Let P = (4, -7) = (p1, p2) and Q = (-1, 5) = (q1, q2).

Then, the components of v = are given by

21,vv

v1 = q1 – p1 =

v2 = q2 – p2 =

-1 – 4 = -5

5 – (-7) = 12

Thus, v = 12,5and the length of v is

1316912)5( 22 v

P

Q

Initial Point

Terminal Point

magnitu

de is th

e length

direction is

this angle

How can we find the magnitude if we have the initial point and the terminal point?

22 , yx

11, yx

The distance formula

How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

22 , yx

11, yx

Q

Terminal Point

direction is

this angle

Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y).

yx,

0,0If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.

P

Initial Point

A vector whose initial point is the origin is called a position vector

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

v w

Initial point of vv

w

Move w over keeping the magnitude and direction the same.

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

Terminal point of w

w

The negative of a vector is just a vector going the opposite way.

v

v

A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

vv

vv3

uv

w vu u

vw3w w

w

Using the vectors shown, find the following:

vu u

v

vwu 32

uu w

w wv

b

av

Vectors are denoted with bold letters

(a, b)

This is the notation for a position vector. This means the point (a, b) is the terminal point and the initial point is the origin.

We use vectors that are only 1 unit long to build position vectors. i is a vector 1 unit long in the x direction and j is a vector 1 unit long in the y direction.

ij

jiv bab

a

(3, 2)

ij

2

3v i i

j

jiv 23

jiji 4352

If we want to add vectors that are in the form ai + bj, we can just add the i components and then the j components.

jiv 52

wv ji

Let's look at this geometrically:

i2

j5 v

i3

j4w

ij

When we want to know the magnitude of the vector (remember this is the length) we denote it

v 22 52 Can you see from this picture how to find the length of v?

29

jiw 43

A unit vector is a vector with magnitude 1.

jiw 43

If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value.

What is ?w

2 23 4 w 525

If we want to find the unit vector having the same direction as w we need to divide w by 5.

jiu5

4

5

3

Let's check this to see if it really is 1 unit long.

2 23 4 25

15 5 25

u

If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form.

5, 150 v

1505

As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

cos sin v v i j

jijiv2

5

2

35150sin150cos5

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au

Vector Operations

The two basic operations are scalar multiplication and vectoraddition. Geometrically, the product of a vector v and a scalark is the vector that is times as long as v. If k is positive, then kv has the same direction as v, and if k is negative, then kv has the opposite direction of v.

k

v ½ v 2v -vv

2

3

Definition of Vector Addition & Scalar Multiplication

Let u = and v = be vectors and let k be a scalar (real number). Then the sum of u and v is

21,uu 21,vv

u + v = 2211 , vuvu

and scalar multiplication of k times u is the vector

2121 ,, kukuuukku

Vector Operations

Ex. Let v = and w = . Find the following vectors. a. 2v b. w – v

5,2 4,3

-2 2

6

10

8

4

2

-2-4

10,42 v

v

2v

1 2 3 4

4

3

2

1

5-1

w

1,554),2(3 vw

-v

w - v

Writing a Linear Combination of Unit Vectors

Let u be the vector with initial point (2, -5) and terminal point (-1, 3). Write u as a linear combination of the standard unitvectors of i and j.

-2 2 4

6

4

2

-2

-4

-6

-8

(2, -5)

u

(-1, 3)

Solution

53,21 u

8,3

ji 83

-2 2

6

10

8

4

2

-2-4

Graphically,it looks like…

-3i

8j

Writing a Linear Combination of Unit Vectors

Let u be the vector with initial point (2, -5) and terminal point(-1, 3).Write u as a linear combination of the standard unitvectors i and j.

Begin by writing the component form of the vector u.

u 1 2,3 5

u 3,8

u 3i 8 j

Unit Vectors

v

v

1

v

vu = unit vector

Find a unit vector in the direction of v =

v

v

2,5

2 2 5 2

2,5

1

29 2,5

2

29,

5

29

Vector Operations

Let u = -3i + 8j and let v = 2i - j. Find 2u - 3v.

2u - 3v = 2(-3i + 8j) - 3(2i - j)

= -6i + 16j - 6i + 3j

= -12i + 19 j

Dot Products

The Definition of the Dot Product of Two Vectors

The dot product of u = and v = is

u1,u2

v1,v2

uv u1v1 u2v2

Ex.’s Find each dot product.

a. 4,5 2,3

b. 2, 1 1,2

c. 0,3 4, 2

4 2 5 3 23

2 1 1 2 0

0 4 3 2 6

Properties of the Dot Product

Let u, v, and w be vectors in the plane or in space and let c be a scalar.

uv v u0v 0

u v w uv uw

v v v 2

c uv cuv ucv

Let

u 1,3 , v 2, 4 , and w 1, 2

Find

uv w First, find u . v

uv 1 2 3 4 14

uv w

14 1, 2 14,28

Find u . 2v = 2(u . v) = 2(-14) = -28

The Angle Between Two Vectors

If is the angle between two nonzero vectors u and v, then

cos uvu v

Find the angle between

u 4,3 & v 3,5

v 3,5

u 4,3

cos 4,3 3,5

4,3 3,5

27

5 34

arccos27

5 3422.2

Definition of Orthogonal Vectors (90 degree angles)

The vectors u and v are orthogonal if u . v = 0

Are the vectors orthogonal?

u 2, 3 & v 6,4

Find the dot product of the two vectors.

uv 2, 3 6,4 2 6 3 4 0

Because the dot product is 0, the two vectors are orthogonal.

End of notes.